Novel TMS coils designed using an inverse boundary element method

Devices and Technology in Transcranial Magnetic Stimulation: A Systematic Review

The technology for transcranial magnetic stimulation (TMS) has significantly changed over the years, with important improvements in the signal generators, the coils, the positioning systems, and the software for modeling, optimization, and therapy planning. In this systematic literature review (SLR), the evolution of each component of TMS technology is presented and analyzed to assess the limitations to overcome. This SLR was carried out following the PRISMA 2020 statement. Published articles of TMS were searched for in four databases (Web of Science, PubMed, Scopus, IEEE). Conference papers and other reviews were excluded. Records were filtered using terms about TMS technology with a semi-automatic software; articles that did not present new technology developments were excluded manually. After this screening, 101 records were included, with 19 articles proposing new stimulator designs (18.8%), 46 presenting or adapting coils (45.5%), 18 proposing systems for coil placement (17.8%), and 43 implementing algorithms for coil optimization (42.6%). The articles were blindly classified by the authors to reduce the risk of bias. However, our results could have been influenced by our research interests, which would affect conclusions for applications in psychiatric and neurological diseases. Our analysis indicates that more emphasis should be placed on optimizing the current technology with a special focus on the experimental validation of models. With this review, we expect to establish the base for future TMS technological developments.

Coil efficiency for inductive peripheral nerve stimulation

Magnetic stimulation of peripheral nerves is evoked by electric field gradients caused by high-intensity, pulsed magnetic fields created from a coil. Currents required for stimulation are very high, therefore devices are large, expensive, and often too complex for many applications like rehabilitation therapy. For repetitive stimulation, coil heating due to power loss poses a further limitation. The geometry of the magnetic coil determines field depth and focality, making it the most important factor that determines the current required for neuronal excitation. However, the comparison between different coil geometries is difficult and depends on the specific application. Especially the distance between nerve and coil plays a crucial role. In this investigation, the electric field distribution of 14 different coil geometries was calculated for a typical peripheral nerve stimulation with a 27mm distance between axon and coil. Coil parameters like field strength and focality were determined with electromagnetic field simulations. In a second analysis, the activating function along the axon was calculated, which quantifies the efficiency of neuronal stimulation. Moreover, coil designs were evaluated concerning power efficacy based on ohmic losses. Our results indicate that power efficacy of magnetic neurostimulation can be improved significantly by up to 40% with optimized coil designs.

Design of Optimal Coils for Deep Transcranial Magnetic Stimulation

In this work, a stream function inverse boundary element method (IBEM) has been used for designing different deep transcranial magnetic stimulation (dTMS)coils to activate the prefrontal cortex and the temporal lobe have been set as the target regions. In addition, the performances of these coils have been described and the electric field induced by them has been obtained by using a computational forward technique. These results show that the stream function IBEM is an ideal approach to design optimal dTMS coils capable of producing deep stimulation in the target brain regions. Clinical relevance - The design problem proposed here can be used to produce efficient dTMS stimulators for neurological disorders, which can overcome some of the currently existing limitations of the most common devices employed in TMS.

Analytical Solutions to Minimum-Norm Problems

For G∈Rm×n and g∈Rm, the minimization min∥Gψ−g∥2, with ψ∈Rn, is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min∥T(h)−k∥, where T:H→K is a continuous linear operator between Hilbert spaces H,K and h∈H,k∈K. In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min∥T(h)−k∥andmin∥h∥. We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min∥T(h)−k∥with∥h∥=α. The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils.

Pareto Optimality for Multioptimization of Continuous Linear Operators

This manuscript determines the set of Pareto optimal solutions of certain multiobjective-optimization problems involving continuous linear operators defined on Banach spaces and Hilbert spaces. These multioptimization problems typically arise in engineering. In order to accomplish our goals, we first characterize, in an abstract setting, the set of Pareto optimal solutions of any multiobjective optimization problem. We then provide sufficient topological conditions to ensure the existence of Pareto optimal solutions. Next, we determine the Pareto optimal solutions of convex max–min problems involving continuous linear operators defined on Banach spaces. We prove that the set of Pareto optimal solutions of a convex max–min of form max∥T(x)∥, min∥x∥ coincides with the set of multiples of supporting vectors of T. Lastly, we apply this result to convex max–min problems in the Hilbert space setting, which also applies to convex max–min problems that arise in the design of truly optimal coils in engineering.

Design of coils for lateralized TMS on mice

OBJECTIVE

Translational studies on animals play a vital role in the advancement of transcranial magnetic stimulation (TMS) as clinical technique. Nonetheless the relevance of these procedures is frequently limited by the lack of TMS systems specifically designed for small animals capable of producing comparable stimulation conditions to those found in human TMS. In this work, we propose to take advantage of the versatility of recently introduced TMS coil design methods to produce optimal rodent-specific TMS stimulators.

APPROACH

A stream function inverse boundary element method (IBEM) has been used for producing three small sized mice-specific TMS coils of different geometries. They have been created for unilateral hemispheric stimulation of the rodent brain, and several constraints have been considered in the design process to satisfy essential performance requirements, such as minimum stored magnetic energy, minimum power dissipation, optimised maximum current density or minimization of the undesired electric field induced in non-target regions. In order to validate the presented strategy, three prototype coils have been built. The performance of each prototype has also been numerically investigated, where the electric field induced in a mouse model has been found by using an existing computational forward technique.

MAIN RESULTS

Stream function IBEM represents an ideally suited approach for designing specific TMS coils for small animals, capable of fulfilling many essential functional and technical requirements. The prototypes produced in this work focally stimulate the right hemisphere of the mouse brain, and so they can be successfully used in lateralized TMS experiments.

SIGNIFICANCE

The design scheme proposed here can be used to produce efficient TMS stimulators for small animals, which can overcome some of the existing limitations found when producing more reliable translational experiments.

Design of TMS coils with reduced Lorentz forces: application to concurrent TMS-fMRI

OBJECTIVE

Interleaving TMS (transcranial magnetic stimulation) with fMRI (functional Magnetic Resonance Imaging) is a promising technique to study functional connectivity in the human brain, but its development is being restricted by technical limitations, such as that due to the interaction of the TMS current pulses with the magnetic fields of an MRI scanner. In this work, a TMS coil design method capable of controlling Lorentz forces experienced by the coil in the presence of static magnetic fields is presented.

APPROACH

The suggested approach is based on an existing inverse boundary element method (IBEM) for TMS coil design, in which new electromagnetic computational models of the Lorentz forces have been included to be controlled in the design process.

MAIN RESULTS

To demonstrate the validity of this technique, it has been used for the design and simulation of TMS coils wound on rectangular flat, spherical and hemispherical surfaces with improved mechanical stability. The obtained results confirm that TMS coils with reduced Lorentz forces inside the static main field of an MRI scanner can be produced, which is achieved to the detriment of other coil performance parameters.

SIGNIFICANCE

The proposed approach provides an efficient tool to design TMS stimulators of a wide range of coil geometries with improved mechanical stability, which can be extremely useful to overcome current limitations for interleaved TMS-fMRI.

Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1

In this manuscript we provide an exact solution to the maxmin problem max ∥ A x ∥ subject to ∥ B x ∥ ≤ 1 , where A and B are real matrices. This problem comes from a remodeling of max ∥ A x ∥ subject to min ∥ B x ∥ , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max ∥ C x ∥ subject to ∥ x ∥ ≤ 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables.

Design of transcranial magnetic stimulation coils with optimal trade-off between depth, focality, and energy

OBJECTIVE

Transcranial magnetic stimulation (TMS) is a noninvasive brain stimulation technique used for research and clinical applications. Existent TMS coils are limited in their precision of spatial targeting (focality), especially for deeper targets. This paper presents a methodology for designing TMS coils to achieve optimal trade-off between the depth and focality of the induced electric field (E-field), as well as the energy required by the coil.

APPROACH

A multi-objective optimization technique is used for computationally designing TMS coils that achieve optimal trade-offs between E-field focality, depth, and energy (fdTMS coils). The fdTMS coil winding(s) maximize focality (minimize the volume of the brain region with E-field above a given threshold) while reaching a target at a specified depth and not exceeding predefined peak E-field strength and required coil energy. Spherical and MRI-derived head models are used to compute the fundamental depth-focality trade-off as well as focality-energy trade-offs for specific target depths.

MAIN RESULTS

Across stimulation target depths of 1.0-3.4 cm from the brain surface, the suprathreshold volume can be theoretically decreased by 42%-55% compared to existing TMS coil designs. The suprathreshold volume of a figure-8 coil can be decreased by 36%, 44%, or 46%, for matched, doubled, or quadrupled energy. For matched focality and energy, the depth of a figure-8 coil can be increased by 22%.

SIGNIFICANCE

Computational design of TMS coils could enable more selective targeting of the induced E-field. The presented results appear to be the first significant advancement in the depth-focality trade-off of TMS coils since the introduction of the figure-8 coil three decades ago, and likely represent the fundamental physical limit.

Redesigning existing transcranial magnetic stimulation coils to reduce energy: application to low field magnetic stimulation

OBJECTIVE

To present a systematic framework and exemplar for the development of a compact and energy-efficient coil that replicates the electric field (E-field) distribution induced by an existing transcranial magnetic stimulation coil.

APPROACH

The E-field generated by a conventional low field magnetic stimulation (LFMS) coil was measured for a spherical head model and simulated in both spherical and realistic head models. Then, using a spherical head model and spatial harmonic decomposition, a spherical-shaped cap coil was synthesized such that its windings conformed to a spherical surface and replicated the E-field on the cortical surface while requiring less energy. A prototype coil was built and electrically characterized. The effect of constraining the windings to the upper half of the head was also explored via an alternative coil design.

MAIN RESULTS

The LFMS E-field distribution resembled that of a large double-cone coil, with a peak field strength around 350 mV m^-1 in the cortex. The E-field distributions of the cap coil designs were validated against the original coil, with mean errors of 1%-3%. The cap coil required as little as 2% of the original coil energy and was significantly smaller in size.

SIGNIFICANCE

The redesigned LFMS coil is substantially smaller and more energy-efficient than the original, improving cost, power consumption, and portability. These improvements could facilitate deployment of LFMS in the clinic and potentially at home. This coil redesign approach can also be applied to other magnetic stimulation paradigms. Finally, the anatomically-accurate E-field simulation of LFMS can be used to interpret clinical LFMS data.